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\headerbox{0.2 Mathematical Notions and Terminology}

\large

Define each of these terms:

\bigskip \indent
\begin{itemize}
\item set
  \subitem An unordered collection of unique objects, called \term{element}s.  Sets are usually designated by their \term{element}s listed within curly braces, such as $A=\set{a,b,c}$.
\item element(s)
  \subitem Any particular object contained in a \term{set} or \term{sequence}.
  Also known as \term{member}.
\item member(s)
  \subitem Any particular object contained in a \term{set}.
  Also known as \term{element}.
\item subset
  \subitem A \term{set} which does not include any \term{element}s that are not included in another \term{set}, known as the \term{superset}.  Thus, if \term{A} is a subset of \term{B}, then there are no elements in \term{A} that are not found in \term{B}.
\item proper subset
  \subitem A \term{subset} which does not include all the \term{element}s of its \term{superset}.  That is, a \term{proper subset} will have at least one \term{element} less than its \term{superset}.
\item multiset
  \subitem An unordered collection of \term{element}s which may or may not be repeated.
\item infinite set
  \subitem A \term{set} which contains an infinite number of \term{element}s.
\item natural numbers $\mathbb{N}$
  \subitem Generally refers to the positive \term{integer}s, $\set{1,2,3,\ldots}$, though in some contexts it is the non-negative \term{integer}s, $\set{0,1,2, \ldots}$.
\item integers $\mathbb{Z}$
  \subitem The positive and negative whole numbers and zero, $\set{\ldots,-2,-1,0,1,2,\ldots}$.  This is equivalent to the \term{natural number}s, their opposites ({\ie} negatives), and zero.
\item empty set $\varnothing$
  \subitem The \term{set} which has no \term{element}s.  Often marked $\varnothing$, but sometimes denoted $\set{}$.
\item union $\cup$
  \subitem A \term{set operation} which generates a new \term{set} containing all the \term{element}s of each source \term{set}.
\item intersection $\cap$
  \subitem A \term{set operation} which generates a new \term{set} containing only the \term{element}s which are found in each of the source \term{set}s.
\item complement $\overline{A}$
  \subitem A \term{set operation}, which generates a new \term{set} containing all the \term{element}s in the universe of discourse which are \emph{not} found in the source \term{set}.  Frequently denoted by a line over the source set, thus the complement of $A$ is $\overline{A}$.
\item compliment

\item Venn diagram

\item sequence

\item tuple(s)

\item k-tuple

\item pair

\item power set %% $\mathcal{P}\left(A\right)$

\item Cartesian product $A \times B$

\item cross product $A \times B$

\item function

\item mapping

\item domain

\item codomain

\item range

\item onto

\item surjective

\item surjection

\item one-to-one

\item injective

\item injection

\end{itemize}

\bigskip \indent
\begin{itemize}
\item 1-to-1 correspondence

\item bijective

\item bijection

\item arguments

\item k-ary function

\item arity

\item unary function

\item binary function

\item infix notation

\item prefix notation

\item postfix notation

\item predicate/property

\end{itemize}

\bigskip \indent
\begin{itemize}
\item relation $R$

\item k-ary relation

\item k-ary relation on A

\item binary relation

\item equivalence relation

\item reflexive

\item symmetric

\item transitive

\item undirected graph

\end{itemize}

\bigskip \indent
\begin{itemize}
\item graph

\item nodes

\item vertices

\item edges

\item degree

\item labeled graph

\item subgraph

\item path

\item simple path

\item connected

\item cycle

\item simple cycle

\item tree

\item root

\item leaves

\item directed graph

\item outdegree

\item indegree

\item directed path

\item strongly connected

\end{itemize}
   
\bigskip \indent
\begin{itemize}
\item \emph{alphabet} $\Sigma$

\item symbols

\item string over an alphabet

\item length $\abs{w}$

\item empty string $\varepsilon$

\item reverse $w^\mathcal{R}$

\item substring

\item concatenation

\item lexicographic ordering

\item language

\end{itemize}

\bigskip \indent
\begin{itemize}
\item Boolean logic

\item Boolean values

\item Boolean operations

\item negation/NOT $\neg$

\item conjunction/AND $\wedge$

\item disjunction/OR $\vee$

\item exclusive or/XOR $\oplus$

\item equality (operation) $\leftrightarrow$

\item implication (operation) $\rightarrow$

\item operands

\item distributive law for AND and OR

\end{itemize}

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